Numerical solution of a 1-D elastohydrodynamic problem in magnetic storage devices

Arregui, Iñigo; Cendán, José Jesús; Parés, Carlos; Vázquez, Carlos

ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 4, pp. 645-665.

In this work we present new numerical methods to simulate the mechanics of head-tape magnetic storage devices. The elastohydrodynamic problem is formulated in terms of a coupled system which is governed by a nonlinear compressible Reynolds equation for the air pressure over the head, and a rod model for the tape displacement. A fixed point algorithm between the solutions of the elastic and hydrodynamic problems is proposed. For the nonlinear Reynolds equation, a characteristics method and a duality algorithm are developed to cope with the convection dominating and nonlinear diffusion features, respectively. Furthermore, in the duality method the convergence and optimal choice of the parameters are analyzed. At each fixed point iteration, in the elastic model a complementarity formulation is required and appropriate numerical techniques are used. For the spatial discretization different finite element spaces are chosen. Finally, numerical test examples illustrate the theoretical results, as well as the good performance in the simulation of real devices.

[1] I. Arregui and C. Vázquez, Finite element solution of a Reynolds-Koiter coupled problem for the elastic journal bearing. Comput. Meth. Appl. Mech. Engrg. 190 (2001) 2051-2062. | Zbl

[2] I. Arregui, J.J. Cendán and C. Vázquez, A duality method for the compressible Reynolds equation. Application to simulation of read/write processes in magnetic storage devices. J. Comput. Appl. Math. 175 (2005) 31-40. | MR | Zbl

[3] I. Arregui, J.J. Cendán and C. Vázquez, Numerical simulation of head-tape magnetic reading devices by a new two dimensional model. Finite Elem. Anal. Des. 43 (2007) 311-320. | MR

[4] A. Bermúdez, Un método numérico para la resolución de ecuaciones con varios términos no lineales. Aplicación a un problema de flujo de gas en un conducto. Rev. R. Acad. Cienc. Exactas Fís. Nat. 78 (1981) 89-96.

[5] A. Bermúdez and C. Moreno, Duality methods for solving variational inequalities. Comput. Math. Appl. 7 (1981) 43-58. | MR | Zbl

[6] B. Bhushan, Tribology and Mechanics of Magnetic Storage Devices. Springer, New York (1996).

[7] G. Buscaglia and M. Jai, A new numerical scheme for non uniform homogenized problems: application to the nonlinear Reynolds compressible equation. Math. Probl. Engrg. 7 (2001) 355-378. | MR | Zbl

[8] G. Buscaglia, S. Ciuperca and M. Jai, Existence and uniqueness for several nonlinear elliptic problems arising in lubrication theory. J. Diff. Eq. 1 (2005) 187-215.

[9] P.G. Ciarlet, Introduction à l'Analyse Numérique Matricielle et à l'Optimisation. Masson, Paris (1982). | MR | Zbl

[10] J. Durany, G. García and C. Vázquez, Numerical computation of free boundary problems in elastohydrodynamic lubrication. Appl. Math. Modelling 20 (1996) 104-113. | Zbl

[11] J. Durany, G. García and C. Vázquez, Simulation of a lubricated Hertzian contact problem under imposed load. Finite Elem. Anal. Des. 38 (2002) 645-658. | MR | Zbl

[12] A. Friedman, Mathematics in Industrial Problems, IMA 97. Springer, New York (1994). | MR | Zbl

[13] A. Friedman and B. Hu, Head-media interaction in magnetic recording, Arch. Rational Mech. Anal. 140 (1997) 79-101. | MR | Zbl

[14] A. Friedman and J.I. Tello, Head-media interaction in magnetic recording. J. Diff. Eq. 171 (2001) 443-461. | MR | Zbl

[15] R. Glowinski, J.L. Lions and R. Trémolières, Analyse Numérique des Inéquations Variationnelles. Dunod, Paris (1976). | Zbl

[16] J. Heinrich and S. Wadhwa, Analysis of self-acting bearings: a finite element approach. Tribol. Mech. Magnet. Stor. Syst., STLE SP-21 3 (1986) 152-159.

[17] M. Jai, Homogenization and two-scale convergence of the compressible Reynolds lubrification equation modelling the flying characteristics of a rough magnetic head over a rough rigid-disk surface. RAIRO Modél. Math. Anal. Numér. 29 (1995) 199-233. | Numdam | MR | Zbl

[18] C.A. Lacey and F.E. Talke, A tightly coupled numerical foil bearing solution. IEEE Trans. Magn. 26 (1990) 3039-3043.

[19] C.A. Lacey and F.E. Talke, Measurement and simulation of partial contact at the head-tape interface. ASME J. Tribol. 114 (1992) 646-652.

[20] C. Parés, J. Macías and M. Castro, Duality methods with an automatic choice of parameters. Application to shallow water equations in conservative form. Numer. Math. 89 (2001) 161-189. | MR | Zbl

[21] C. Parés, J. Macías and M. Castro, On the convergence of the Bermúdez-Moreno algorithm with constant parameters. Numer. Math. 92 (2002) 113-128. | MR | Zbl

[22] J.N. Reddy, An Introduction to Finite Element Methods. McGraw-Hill (1993). | Zbl

[23] K.J. Stahl, J.W. White and K.L. Deckert, Dynamic response of self acting foil bearings. IBM J. Res. Dev. 18 (1974) 513-520.

[24] S. Tan and F.E. Talke, Numerical and experimental investigations of head-tape interface in a digital linear tape drive. ASME J. Tribol. 123 (2001) 343-349.

[25] S.R. Wu and J.T. Oden, A note on some mathematical studies in elastohydrodynamic lubrication. Int. J. Eng. Sci. 25 (1987) 681-690. | MR | Zbl

[26] Y. Wu and F.E. Talke, Design of a head-tape interface for ultra low flying. IEEE Trans. Magn. 32 (1996) 160-165.

[27] Y. Wu and F.E. Talke, Finite element based head-tape interface simulation including head-tape surface asperity contacts. IEEE Trans. Magn. 34 (1998) 1783-1785.

[28] Y. Wu and F.E. Talke, A finite element simulation of the two-dimensional head-tape interface for head contours with longitudinal bleed slots. Tribol. Int. 22 (2000) 123-130.